# Equations Involving Absolute Values

Example 1. Solve $|2x-7|=3$.

If $|a|=3$ then either a=3 or a=-3. This means that given equation is equivalent to the set of equations: $2x-7=3,2x-7=-3$.

From first equation we find that x=5, from second we find that x=2. Thus, there are 2 solutions: 2 and 5.

Example 2. Solve $|2x-8|=3x+1$.

We need to consider two cases: $2x-8>=0$ and $2x-8<0$.

If $2x-8>=0$ then $|2x-8|=2x-8$ and given equation can be rewritten as $2x-8=3x+1$. From this we have that x=-9. However, when x=-9 inequality $2x-8>=0$ doesn't hold ($2*(-9)-8=-26<0$), therefore, x=-9 is not root of the equation.

If $2x-8<0$ then $|2x-8|=-(2x-8)$ and given equation can be rewritten as $-(2x-8)=3x+1$. From this we have that $x=7/5$ . When $x=7/5$ inequality $2x-8<0$ holds ($2*(7/5)-8=-26/5<0$), therefore, $x=7/5$ is root of the equation.

Therefore, there is only one root: $x=7/5$.

Note, that equation of the form $|x-a|=b$ can be also solved geometrically.